ME 328 – Machine Design
Vibration handout (vibrations is not covered in text)
The following are two good textbooks for vibrations (any edition). There are numerous
other texts of equal quality.
M. L. James, et al., Vibration of Mechanical and Structural Systems, Harper-Collins
College Publishers.
R. Vierck, Vibration Analysis, Harper and Row.
The following is an outline of vibration subjects covered in ME328. Only single degree
of freedom will be investigated:
 Basic concepts of vibrations and waves
o What are the basic features of harmonic motion?
 Analysis of simple harmonic motion of undamped free vibration
o What is the natural frequency?
 Basic behavior of viscous damped free vibration.
o What is the qualitative behavior of a damped system?
 Analysis of forced vibration without dampening.
o What is the vibration amplitude if an undamped system is excited with
force input?
 Analysis of forced vibration with viscous damping.
o What is the vibration amplitude if a damped system is excited with force
input?
 Isolation and transmission of vibration forces of forced vibration with viscous
damping.
o How much force is transmitted from the vibrating system to the
surrounding support structure?
ME 328 – Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 16
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VIBRATION BASICS
In this section we will study basic features of harmonic motion.
Why is vibration worth studying? What effects does it have (good and bad)?
What is meant by “degree of freedom” (DOF)?
Sketch a 1 DOF system:
How many degrees of freedom does a rigid body have, and what are they?
Sketch a 2 DOF system:
How many degrees of freedom does a real structure have, say for example a simple
cantilever beam or a guitar string?
What is meant by “mode”?
How many modes were apparent in the Tacoma Narrows Bridge? Describe them.
What is meant by “fundamental frequency”?
ME 328 – Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 16
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What are “overtones”?
Waves:
Vibration is a oscillating motion, therefore it is essential to define basic concepts of
sinusoidal waves.
x(t) = A sin(t + )
x(t) = displacement or amplitude as a function of time (meters)
t = time (seconds)
 = angular frequency (radians per second)
 = phase angle
A = peak amplitude (meters)
x(t)
time


 = period, how much time per cycle
f = frequency, cycles per second (1 Hz = 1 cycle/sec)
f = 1/
 = 2 f
ME 328 – Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 16
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SIMPLE HARMONIC MOTION OF UNDAMPED FREE VIBRATION
In this section, the main objective is to determine natural frequency of a system.
Simple mass-spring system at equilibrium (not moving):
+x
equilibrium
-x
If a system is displaced from equilibrium it will move to seek out equilibrium. If there is
no damping, the system will continually convert energy back and forth between kinetic
energy of the moving mass and elastic energy of the spring.
x(t)
+x
+x
-x
-x
time
Assumptions we will make for analysis:
1 Degree of freedom, which requires (why, how will system behave differently if
assumptions are not valid?):
Rigid support
Massless, linear elastic spring
Displacement of mass is in one direction only
Mass is rigid
No losses (no damping, etc.)
ME 328 – Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 16
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k = spring stiffness (lb/in, N/m, etc.)
n = natural angular frequency = 2/n (radians/second)
fn = natural frequency = 1/n (cycles/second, Hz)
What is natural frequency?
How will the mass affect natural frequency? Will increased mass increase the natural
frequency or decrease it?
How will spring stiffness affect natural frequency? Will increased stiffness increase the
natural frequency or decrease it?
Sketch FBD & kinetic diagram:
Solutions to equations of motion:
x = A sin(nt)
displacement as a function of time

x = An cos(nt)
velocity as a function of time

x = -An2 sin(nt)
acceleration as a function of time
Equilibrium equation:
inertial forces = - elastic forces (momentum opposes the elastic restoring force)

m x = - kx = m(-An2 sin(nt)) = - k (A sin(nt))
Solving the equation gives:
n2 = k/m; n =
k /m
Eq. 1
Does this make sense? Did you expect an increase in stiffness to increase the natural
frequency? Did you expect an increase in mass to decrease the natural frequency?
ME 328 – Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 16
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If the mass-spring system was rotated and placed on frictionless rollers such that gravity
is perpendicular to the direction of motion,
a) How would natural frequency change?
equilibrium
b) How would amplitude (A) of oscillation change?
c) How would the equilibrium position change?
Example
Given:
An elevator car with mass of 4000kg
Suspended by a steel cable, 20m long, 6cm2 cross-section
Find:
Natural frequency of the system
Assume:
Sketch:
Solution:
ME 328 – Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 16
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Example
Given:
An aluminum cantilever beam with cross section of 10cm by 10cm
A 100kg reciprocating single piston air compressor is placed at the end of the
beam
The compressor spins at 3600RPM
Find:
Beam length such that n=f
Assumptions:
Sketch:
Solution:
ME 328 – Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 16
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ME 328 – Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 16
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VISCOUS DAMPED FREE VIBRATION
(TRANSIENT CONDITION)
The main objective of this section is to understanding damping. We will focus on
qualitative behavior, not quantitative. This is the only condition we will look at where
the motion is transient (non-steady state).
What typically creates viscous damping?
What is the damping force proportional to for viscous damping?
Are there other types of damping?
What is the force proportional to for friction damping?
Will we study other forms of damping in ME328? No.
Tidbits of shock absorber history:
1901 – first patent for hydraulic shock absorber
Oil
Piston
1925 – hydraulic shocks used on automobiles
Shaft
Holes in piston allow oil to flow from one side to the other (viscous forces are produced).
Various relationships between velocity and force can be created by varying design:
progressive
linear
Force
digressive
Velocity
Progressive and digressive shock absorbers can be created by using shims (flaps) to
partially cover the passage holes in the piston giving a non-linear response. This can also
be used to create a different response depending on the direction of travel (up verses
down for example). We will only consider linear response (force is directly proportional
to velocity).
ME 328 – Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 16
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gas
Gas shock absorbers use gas to pressurize the oil
(typically nitrogen). What two effects does this have?
Oil
Piston
Shaft
The system we will study consists of a mass, spring, and dashpot (shock absorber). The
dashpot is viscous (force is proportional to velocity).
Sketch FBD & kinetic diagram
k
c
+x
equilibrium
-x
k = spring constant (as before)
c = damping constant
Equilibrium equation:
acceleration force = - elastic force - damping force




m x = - kx - c x dividing by mass gives: x + c/m x + k/m x = 0
The critical damping constant, cc is:
cc = 2mn
Eq. 2
The damping factor, , is:
 = c/cc
Eq. 3
What is critical damping? (“critical” is not the same as “very important”)
What are the units of the damping constant, c?
ME 328 – Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 16
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If a mass, spring, dashpot system is displaced from equilibrium and released (zero initial
velocity), on the same graph sketch the response for:
a)  < 1 (under-damped) (Zeta)
b)  = 1 (critically damped)
c)  >1 (over-damped)
x(t)
time
If a critically damped system is displaced from equilibrium and released with the
following initial velocities, sketch the response

a) x > 0 (velocity away from equilibrium position)

b) x = 0 (released without initial velocity)

c) x < 0 (velocity towards equilibrium position)
x(t)
time
ME 328 – Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 16
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ANALYSIS OF FORCED VIBRATION WITHOUT DAMPENING
(STEADY STATE CONDITION)
In this section we will study the effect of harmonic forced input (an external force
applied sinusoidally). We are primarily interested in determining the maximum
displacement of the system.
What are some potential sources of harmonic forces?
Sketch FBD & kinetic diagram
+x
equilibrium
-x
P(t) = P0 sin(f t)
P(t) is a harmonic force input. Question: what is f? How is it different than n?
P is the harmonic force. It produces a “push” and a “pull” in an oscillatory manner with
an angular frequency of f. f is sometimes referred to as the forcing frequency.
Equilibrium equation:

m x = - kx + P(t) = -kx + P0 sin(f t)
Solving:
x(t) =
P0
sin(f t) =
k  m 2f
P0 / k
1

2
f
sin(f t) =
k /m
P0 / k



1   f

n

2
sin(f t)
Eq. 4
Let X0 = P0/k
X0 is the static displacement due to the static force P0.
Eq. 5
Let r = f / n
r is known as the frequency ratio.
Eq. 6
In this class, we are NOT concerned with x(t), but we are concerned with
amplitudes.
ME 328 – Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 16
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Substituting X0 and r into the previous equation for x(t) gives:
x(t) =
X0
sin(f t)
1  r2
Now let
X=
Eq. 7
X0
1  r2
Eq. 8
X is the peak amplitude of displacement it is not a function of time. Finally,
x(t) = X sin(f t)
Eq. 9
Three conditions may exist;
r < 1 (forcing frequency is less than the natural frequency; f < n)
r > 1 (forcing frequency is greater than the natural frequency; f > n)
r = 1 (forcing frequency is equal to the natural frequency; f = n)
If r < 1
P(t)
time
x(t)
time
P(t)
time
x(t)
time
sketch x(t)
If r > 1
sketch x(t)
How is the phase
different than for
r < 1?
ME 328 – Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 16
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If r = 1
P(t)
time
x(t)
time
Then:
x (t )  
X o f t
2
cos  f t
An important question to answer is: What is the amplitude of the actual vibration?
The magnification factor, MF, is the ratio of the actual vibration amplitude (X)
normalized by the displacement (X0). In all cases MF is a function of the frequency ratio,
r. Remember r =
x (t ) 
f
. From Equation 7, the vibration without damping is:
n
Xo
sin ( f t ) ; where X0 = P0/k
1 r2
By definition of magnification factor (ratio of X to X0):
MF 
X0
X
; where X =
(per Equation 8)
Xo
1  r2
Then:
MF 
1
(MF is always positive so absolute values are used)
|1  r 2 |
Eq. 10
NOTE: MF is NOT a function of time.
X = dynamic peak displacement (magnitude of vibration displacement).
Xo = static displacement created by static force of magnitude P0.
Both X and X0 are magnitudes, they do not vary with time.
What do you expect the magnification factor to be? Should it always be greater than 1,
less than 1, or will it sometime be greater than 1 and sometimes less than 1?
ME 328 – Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 16
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Notice X0 is not a function of r (f or n) but that X is a function of r. Therefore, MF is a
function of the frequency ratio, r. What happens (what is the vibration amplitude) at:
Frequency ratio,
r = f / n
Static
displacement, X0
r = 0 (static
load); (f = 0)
X0
r = 1 (f = n)
X0
r < 1 (f < n)
X0
r = 21/2
X0
r >>1 (f >>n)
X0
Actual
displacement
amplitude, X
Magnification
Factor, MF
Sketch MF vs. r
3
MF
2
1
0
1
21/2
r
2
ME 328 – Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 16
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ANALYSIS OF FORCED VIBRATION WITH VISCOUS DAMPING
(STEADY STATE CONDITION)
The prior section determined the magnification factor for undamped systems. The
objective of this section is to determine the magnification factor when viscous damping
is present. We will also determine the maximum magnification factor of a given
system.
Sketch FBD & kinetic diagram:
k
c
+x
equilibrium
-x
P(t) = P0 sin(f t)
Equilibrium equation:
mx   kx  cx  P(t )
The magnification factor for this condition is:
MF 
1
X

Xo
Eq. 11
(1  r )  ( 2 r ) 2
2 2
NOTE: magnification factor is NOT a function of time, it is a constant assuming
steady state conditions. As previously defined:
 
c
cc
r
f
n
And: cc  2mn
3
Sketch MF for:
a) no damping
b)  = 0.2
c)  = 0.707
MF
2
1
0
1/2
2
2
1
ME 328 – Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 16
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16
Show that the magnification factor defined in Equation 11, becomes the equation defined
in Equation 10 if there is no damping:
X
1

Equation 11: MF 
Xo
(1  r 2 ) 2  ( 2 r ) 2
Equation 10: MF 
1
|1  r 2 |
By using the magnification factor (MF) we can determine the amplitude of vibration for a
forced vibration system with or without viscous damping. In real systems, mass, spring
constant and damping constant typically do not vary but the forcing frequency (f) will
change. For example, your car has a certain mass, springs, and shock absorbers, but the
engine runs at different speeds. Therefore, the next question we may want to answer is
for a given system (fixed mass, spring, and dashpot), what is the maximum magnification
factor (MFmax) for any forcing frequency (f)?
For a damping factor greater than 0.707 (  
less than 1 for all values of r.
However, for  
MFmax 
1
2
1
) the magnification factor will remain
2
Eq. 12
the maximum magnification factor (MFmax) is:
X max

Xo
2
1
1  2
Eq. 13
The last question we will answer is at what frequency ratio (r) will the magnification
factor be maximized? The answer depends upon the damping factor:
1
2
The magnification factor is maximum at r = 0 (MFmax = 1 and occurs at rMFmax = 0).
Therefore, with any non-zero value of r, the dynamic response (vibration) will be less
than the static displacement due to a static force of P0 and will decrease with
increasing f.
for  
ME 328 – Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 16
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for  
1
2
The magnification factor will be maximized at the frequency ratio of rMFmax:
rMF max  1  2
2
Eq. 14
To summarize, the magnification factor (MF) is the ratio of the displacement amplitude
of the vibrating system to the displacement of the system due to a static load of
magnitude P0. It is a function of the system characteristics (mass, spring, dashpot) and
the forcing frequency (f). For a given system (with constant mass, spring, dashpot), we
can determine the maximum magnification factor for all forcing frequencies (MFmax) and
we can determine the frequency that maximizes the magnification factor (rMFmax).
ME 328 – Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 16
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ISOLATION AND TRANSMISSION OF VIBRATION FORCES OF FORCED
VIBRATION WITH VISCOUS DAMPING
(STEADY STATE CONDITION)
So far we have analyzed the natural frequency of a mass-spring system and determined
the vibration amplitude as a function of mass (m), spring (k), dashpot (c), and forcing
frequency (f ) for a forced vibration system with or without damping. The last
question we will answer is what is “what is the maximum force transmitted from the
vibrating system to the structure holding it in place?”
We will consider forced vibration with viscous damping. Although there is damping, the
system will continue to oscillate indefinitely due to the external force, P. We will
investigate the force transmitted by this steady state oscillation.
k
c
+x
equilibrium
-x
P = P0 sin(f t)
In addition to the inertial force, there are three forces acting on the mass: the spring force
(proportional to displacement), the force from the dashpot (proportional to velocity) and
the external force, P. All of these forces are a function of time. The free body diagram
(FBD) is:
kx
c x
Plus:
P = P0 sin(f t)
The equilibrium equation is:
mx  kx  cx   Po sin ( w f t )
The inertial force (m x ) and the external force (P) are reacted by the spring and dashpot.
Only the spring and dashpot are attached to the surrounding structure, and therefore the
sum of these two forces is the transmitted force:
ME 328 – Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 16
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Ftransmitted  kx  cx
Eq. 15
Although the transmitted force will vary with time (harmonically), we are interested only
in the maximum value. Let FT be the amplitude of the transmitted force (it does NOT
vary with time, it is a constant). Similarly to the magnification factor (MF), we will
normalize the transmitted force amplitude by the static force, P0. This ratio is the so
called transmissibility ratio (TR):
Transmissibility ratio: TR 
FT
Po
NOTE: this is NOT a function of time.
We can express the transmissibility ratio as a function of the system’s characteristics, m,
k, c, and f:
F
TR  T 
Po
k 2  ( c f ) 2
Eq. 16
( k  m f ) 2  ( c  f ) 2
2
We can also express it in terms of the normalized characteristics,  and r:
TR 
1  ( 2r ) 2
FT

Po
(1  r 2 ) 2  ( 2r ) 2
Eq. 17
Sketch the transmissibility ratio as a function of frequency ratio for:
a)  = 0
b)  = 0.5
3
TR
2
1
0
1
21/2
r
2
Note that for all values of damping (), the transmissibility ratio equals one (TR=1) at
both r = 0 and at r  2 . The maximum transmitted force will always occur at f < n
regardless of the system characteristics (m, k, c). The frequency ratio for maximizing the
transmitted force, is rTRmax and is given by:
ME 328 – Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 16
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TRmax occurs at:
rTR max 
(1  8 2 )1 / 2  1
2
(this is always less than 1). Eq. 18
Example:
Given: An air compressor is placed on an isolation table with spring and dashpots on all
4 corners
The total mass of table and compressor is 800 kg.
Springs: k = 60 KN/m
Dashpot (shock absorbers): c = 4000 kg/sec
Forcing function is given as P(t) = Posin(f t)
where f = 370 rad/sec (3600 rpm) and Po = 500N
Find: a) Damping ratio, 
b) What forcing frequency (f ) results in maximizing the displacement?
c) Determine the vibration displacement amplitude at the forcing frequency in
part b.
d) What forcing frequency (f ) results in maximizing the transmitted force?
e) Determine the transmitted force at the forcing frequency in part d.
Assume:
Sketch:
Solution:
ME 328 – Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 16
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ME 328 – Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 16
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CONCLUSION & SUMMARY
We have studied the following systems for one degree of freedom:

Analysis of simple harmonic motion of undamped free vibration
o The natural frequency if a function of mass and spring constant:
o n = k / m

Basic behavior of of viscous damped free vibration system.
o When a damped system is displaced from equilibrium, it will seek out
equilibrium and eventual will come to rest at equilibrium (unless acted
upon by a varying external force).
o If it is under-damped, it will oscillate before coming to rest.
o If it is critically damped, it will not oscillate (if initial velocity is zero) but
rather it will approach equilibrium asymptotically.
o If it is over-damped, it will not oscillate (if initial velocity is zero) and will
take a longer time to reach equilibrium than a critically damped system.

Analysis of forced vibration without dampening.
o The vibration amplitude (X) of forced vibration without dampening is a
function of the system characteristics (m and k) as well as the forcing
frequency (f). The ratio of the amplitude normalized by the
displacement created by a static force, P0 is called the magnification
factor, MF. Although the system oscillates (x(t)), the magnification factor
is based on the amplitudes and, therefore, is constant for the system as
long as the forcing frequency (f) remains constant.
X
1

o MF 
X0
|1  r 2 |

Analysis of forced vibration with viscous damping.
o The magnification factor for a system with viscous damping is defined in
the same way as for no damping, but the equation is more complex.
X
1

o MF 
2
2
Xo
(1  r )  ( 2 r ) 2
o For < 0.707, the magnification factor will remain less than 1 for all
values of r.
o For  < 0.707, the maximum magnification factor for a given system (m,
k, c) for all frequency ratios (r) is:
X
1
MFmax  max 
Xo
2 1   2
And occurs at: rMF max  1  2
2
ME 328 – Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 16
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
Isolation and transmission of vibration forces of forced vibration with
viscous damping.
o The amplitude of the force transmitted from the vibrating system to the
surrounding support structure is FT. By normalizing FT by the static force,
P0, the transmissibility ratio, TR, is described as:
F
TR  T 
Po
k 2  ( c f ) 2
( k  m f ) 2  ( c  f ) 2
2

1  ( 2r ) 2
(1  r 2 ) 2  ( 2r ) 2
o Transmissibility ratio does NOT vary with time; it is a constant for a given
system and forcing frequency. Even though there is viscous damping, the
system is assumed to have steady state oscillation due to harmonic
external force, P.
o For a given system (m, k, c) the transmissibility ratio is a function of the
forcing frequency, and it becomes maximum at the frequency ratio of
rTRmax:
rTR max 
(1  8 2 )1 / 2  1
2
o rTRmax is always less than 1 regardless of system characteristics (m, k, c).
In other words, the maximum transmitted forces for a system will always
occur when the forcing frequency is less than the natural frequency.
ME 328 – Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 16
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Magnification Factors
for various damping factors
Magnification Factor (MF)
5
4
z=0
3
z=0.1
z=2
2
z=0.5
z=10
z=1
1
0
0
0.5
1
1.5
2
2.5
Frequency ratio (r)
MF 
1
X

Xo
(1  r )  ( 2 r ) 2
2 2
Transmissibility Ratio
(TR)
5
Transmissibility ratio
for various damping factors


4

3
2


1
0
0
0.5
1
1.5
2
2.5
Frequency ratio (r)
TR 
1  ( 2r ) 2
(1  r 2 ) 2  2r ) 2
ME 328 – Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 16
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